Some of the issues raised in question 1 below are addressed in the
volume: Model Theoretic Logics, edited by Feferman and Barwise. It was
published by Springer Verlag in the 80´´s.
On Wed, October 4, 2006 6:48 pm, Erik Douglas wrote:
>> I have two questions for the erudite members of this list from the
> foundations of logic. Some of the answers may extend outside the range of
> topics appropriate to this list, but I would be very grateful all the same
> for every response; I am a philosopher by trade, and so I am fearless if
> not
> also a bit foolish. My email is erik at temporality.org and I will also
> eventually compile the answers in an informal discussion on the web (and
> possibly more).
>> 1. What are the qualifications on a (formal) language to be a logic?
>> In general, a logic appears to belong to a subclass of formalized
> languages,
> in a larger category of symbolic systems used to communicate. I do not
> mean
> to ask the rather obscure philosophical questions about what it means to
> communicate, or even in what symbolic systems consist. Rather, I accept
> languages as primitive, and moreover that some are *formal* (explicit and
> unambiguous in their application). Also, it seems important that such
> formal languages have both syntactic and semantic dimensions, that we can
> identify something like *expressions* in the syntax, as well as some
> minimal
> distinction between *semantic* kinds amongst those expressions (e.g., TRUE
> and FALSE).
>> The question is, given these minimal criteria, what further conditions
> qualify, generally, a particular formal language as a logic? There is a
> surprising paucity in the literature addressing this question (I have
> found
> some references, but I would also be grateful for any and all in your
> responses). At some level, the matter appears to be conventional or
> metaphysical (depending significantly on your religious inclinations).
> However, in either case, I would be very grateful to know your operative
> (and/or cherished) beliefs here.
>> The second question (below) then turns on what is often assumed as such a
> fundamental condition, the law of non-contradiction. It seems other laws
> that were at one time accepted as etched in stone, such as the law of
> excluded middles, have given way to intuitionist and fuzzy logics. In
> recent years, we have seen the construction of several variety of
> *paraconsistent* logics which do prima facie deny the law of
> non-contradiction. So my question is as follows, and I have in mind that
> many of the folks on this list are mathematicians:
>> 2. Suppose I present a paraconsistent logic that I claim suffices as a
> foundation for mathematics (or some significant part thereof), what
> (other)
> properties must it have for you to accept it as such?
>> Note on my motivation:
>> I am in the business of attempting to construct alternative logics,
> notably
> paraconsistent logics, with an eye to addressing certain issues that sit
> uneasily these days between philosophy, metaphysics, mathematics, and
> physics, especially those that turn on how we understand *time* and
> *mind*.
> There are other applications in the development of a universal logic that
> also motivate my investigation here.
>> Cheers,
> Erik
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